Use the Theorem of Pappus to find the exact volume of a general cone created by rotating a triangle with vertices ( 0 , 0 ) , ( 1 , 0 ) , and ( 0 , 8 ) around the y -axis.

The exact volume of a general Cone is [tex]\frac{8\pi}{3}[/tex]

To solve this problem, we will make use of the second theorem of Pappus. This theorem is concerned with finding the volume produced when a plane shape is rotated around an axis ( the volume of the solid of revolution ).

The theorem states:

The volume of a solid of revolution obtained by rotating a lamina F (the triangle) about a non-intersecting axis (the y-axis) lying in the same plane is equal to the product of the area A of the lamina F and the distance d traveled by the centroid of F.

That is

[tex]V=A\times d[/tex]

When the centroid travels, it forms a circumference d given by,

[tex]d=2\pi r[/tex]

Therefore

[tex]V=A\times 2\pi r[/tex]

From the diagram below,

The vertices (or, corners) of the triangle are A(0, 0), B(1, 0), and C(0,8).

G is the centroid of the triangle. The centroid of the triangle is the geometric center of the triangle. The centroid would be the balance point if a pin were to be used to balance the triangle.

r is the distance from the centroid to the axis of rotation ( the y-axis in this case). It is like saying the radius of rotation.

The things we need to find are A (the Area) and r.

Calculate A:

[tex]A=\frac{1}{2} \left| \Delta \right|[/tex]

          [tex]\Delta = \left|\begin{array}{cc} x_{B}-x_{A} & y_{B}-y_{A} \\ x_{C}-x_{A} &y_{C}-y_{A}\end{array}\right|= \left|\begin{array}{cc} 1-0 & 0-0 \\ 0-0 & 8-0\end{array}\right|= \left|\begin{array}{cc} 1 & 0 \\ 0 & 8\end{array}\right|[/tex]

          [tex]\Delta =8\times 1 -0\times 0=8[/tex]

[tex]A=\frac{1}{2} \left|8\right|=\frac{1}{2}\times 8=4[/tex]

Calculate r:

[tex]r=\overline{x}=\frac{x_A+x_B+x_C}{3}=\frac{0+1+0}{3}=\frac{1}{3}[/tex]

Finally, Calculate the volume, V:

[tex]V=A\times 2\pi r=4 \times2\pi\times\frac{1}{3}=\frac{8\pi}{3}[/tex]

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